2024-03-26 Q1(m=6)

Rearrange the digits in ⟨1263045⟩ to meet the rules below.

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨ ⁶ᵗʰa   ⁴ᵗʰb   ²ⁿᵈc     ⟩, a > b > c

1st → a, 0th → b, a+b=1+6n

2nd → a, 0th → b, a+b=3

⟦0,1⟧ ∋ 2,4

⟨       ³ʳᵈ↓       ⟩ after ⟨→⟩

#125034_v2.4

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │   │   │   │   │   │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │   │   │   │   │ 2 │   │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │   │   │   │   │ 2 │ 6 │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │   │ 0 │   │   │ 2 │ 6 │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │   │ 0 │ 3 │   │ 2 │ 6 │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │   │ 0 │ 3 │ 4 │ 2 │ 6 │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 5 │ 0 │ 3 │ 4 │ 2 │ 6 │ 1 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

Proof of 2024-03-26 Q1(m=6)
═══════════════════════════

Notation: if nth -> a, then we write [nth] = a.

By ✅「1st → a, 0th → b, a+b=1+6n」, we have

(1) [1st] + [0th] = 1 | 7.

If the above sum is 1, then { [1st], [0th] } = {0,1}, but then we cannot match ✅「⟦0,1⟧ ∋ 2,4」, which is a contradiction. It follows that

(2) [1st] + [0th] = 7.

On the other hand, by ✅「2nd → a, 0th → b, a+b=3」, we have

(3) { [2nd], [0th] } = {0,3} | {1,2}.

A fortiori, [0th] = 0|1|2|3. Observe that

(i) To satisfy (2), we need [0th] != 0;

(ii) ✅「⟦0,1⟧ ∋ 2,4」 implies that 2 is not in corners, so [0th] != 2;

(iii) If [0th] = 3, then (2) gives [1st] = 4, but then we cannot match ✅「⟦0,1⟧ ∋ 2,4」:

┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│ 1▲│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│   │   │   │   │   │ 4 │ 3 │
└───┴───┴───┴───┴───┴───┴───┘

Therefore, we have [0th] = 1. Then, by ✅「2nd → a, 0th → b, a+b=3」 and (2), we get [2nd] = 2 and [1st] = 6 as well:

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│ 2■│ 1■│ 0■│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │   │   │   │   │   │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │   │   │   │   │ 2 │   │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │   │   │   │   │ 2 │ 6 │ 1 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │   │ 3 │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Next, we consider where to place 0. It is the minimum digit, so by ✅「⟨       ³ʳᵈ↓       ⟩ after ⟨→⟩」 it cannot be at 3rd. Similarly, by ✅「⟨ ⁶ᵗʰa   ⁴ᵗʰb   ²ⁿᵈc     ⟩, a > b > c」, it cannot be at 6th or 4th. Consequently, 0 = [5th]:

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│ 5■│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │   │   │   │ 2 │ 6 │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │   │ 0 │   │   │ 2 │ 6 │ 1 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │   │ 3 │   │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

We proceed to determine where to place 3. Note that the other idle digits (4 and 5) are both greater than it. Therefore, by ✅「⟨       ³ʳᵈ↓       ⟩ after ⟨→⟩」, we have 3 != [3rd], and by ✅「⟨ ⁶ᵗʰa   ⁴ᵗʰb   ²ⁿᵈc     ⟩, a > b > c」, we have 3 != [6th]. It follows that 3 = [4th]:

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│ 4■│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │ 0 │   │   │ 2 │ 6 │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │   │ 0 │ 3 │   │ 2 │ 6 │ 1 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Finally, noting that there is only one way to match ✅「⟦0,1⟧ ∋ 2,4」, we finish by

       ┌───┬───┬───┬───┬───┬───┬───┐
       │ 6■│5th│4th│ 3■│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │ 0 │ 3 │   │ 2 │ 6 │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │   │ 0 │ 3 │ 4 │ 2 │ 6 │ 1 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 5 │ 0 │ 3 │ 4 │ 2 │ 6 │ 1 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │   │
└───┴───┴───┴───┴───┴───┴───┘

Q.E.D.

#125034_v2.4